Camille O'Sullivan is standing in front of a full-length mirror being fitted into her new costume. "Sweet mother of Jesus," she yelps, as a large safety pin narrowly misses her bosom. When the Irish singer was on stage in Edinburgh in 2009 – the last time I saw her perform – she wore a tight bodice and fishnets, her deep curves and darkly alluring figure straight out of the Weimar Republic. This was for her intense performance of songs by Leonard Cohen, Tom Waits and David Bowie. But today she is being eased into a white shift dress with a round neck and no waistline. "Very simple, very pure," she says. Then comes a sweeping cream coat, fastened at the neck. She is the image of grown-up, womanly dignity.

But then the Irish singer is preparing for a different kind of show: a performance of Shakespeare's poem The Rape of Lucrece, co-created with composer Feargal Murray and theatre director Elizabeth Freestone. After a four-night run in Stratford last year, they are refining the piece for its appearance at the Edinburgh international festival next week. It's a coal-black work: Tarquin, king of Rome, rapes the famously chaste wife of his friend Collatinus; to wipe out the stain of shame, Lucrece kills herself. Tarquin is then banished, an act that will usher in the Roman republic. In the piece, O'Sullivan plays victim and rapist, bereaved husband and father, not to mention the dispassionate narrator.

It was Freestone who challenged O'Sullivan to take on the forbidding 1,855-line poem. Of the singer, she says: "She's comfortable in the territory Lucrece covers – vulnerable enough to get into dark corners, strong enough to come out the other side with grace and humour. She won't hold back: she meets it head on and demands the audience does the same."

O'Sullivan, though a skilled vocalist, has formal training in neither music nor acting. She swapped a career in architecture for one as a performer after a long, soul-searching stint in hospital following a car accident in 1999. Perhaps because she is self-taught, she has carved out her own distinctive style: dangerously committed performances coloured by prewar Berlin cabaret and a childhood spent listening to her French mother's Jacques Brel records.

"My approach to music is quite instinctual and filmic," she says. "I see a song almost like an actress sees a monologue. I see images from my life, from my past. So with Shakespeare, where there is so much metaphor, so much imagery, I feel completely at home. It's in 3D. I feel that, after all these years singing Nick Cave and Jacques Brel songs that have very developed characters, I've actually learned something."

Carving an 85-minute dramatic show out of Shakespeare's narrative was the first challenge. "We had the poem up on the wall and we went through it, deciding what parts could be songs. We really had to get the kernel of what our story was. We had to get rid of a lot of things – beautiful descriptions, some of them."

In an extended sequence, after the rape, Lucrece examines a picture depicting the events and characters of the Trojan war, finding it reflects her own situation. Although it's a virtuosic passage of poetry, it ended up being cut. "You imagine me trying to do that?" says O'Sullivan. "The whole Trojan war? You'd be going, 'Jesus – a five-hour show.'" Instead, they concentrated on moments when characters reveal themselves, the points at which they change. "We were thinking, 'When do they speak, when do they think? What are going to be the moments when the audience will understand that person?'"

Oddly, the result is sometimes reminiscent of baroque opera – with narrated sections like spoken recitative, and aria-like moments when O'Sullivan breaks into song. The sung sections had to seem unforced, though. "Shakespeare the Musical is what we didn't want: tap dancing across the stage singing, 'And then he raped Lucrece!'"

Freestone describes their rehearsal process: "Everything is different from a traditional rehearsal. We don't sit down and look at a page, talk about it, then get it on its feet like on a play. We work on everything – music, meaning, staging – all at the same time. We've learned that a melody can unlock a meaning, a move can make sense of a musical shift, or a repositioned line can be the answer to a tricky song structure. Nothing happens in isolation."

A major challenge has been how O'Sullivan slips between all her characters. "You don't necessarily have to change your physique, or your voice, but there is something in your intent that has to become different," she says. In particular, she has to find what makes both Tarquin and Lucrece tick. Neither of their motivations is obvious: Tarquin seems to commit the rape for the hell of it, and Lucrece's suicide is a deeply ambiguous act that has caused debate among readers from St Augustine onwards (Shakespeare's story is derived from Livy and Ovid). Why would an innocent woman feel so polluted that she had to kill herself?

"It's not just black and white," says O'Sullivan. "It's not just innocence and darkness. You have to show that Tarquin could have been the nice guy; instead he did the worst thing imaginable. He's quite charismatic, capable of charm, and he pretty much regrets the rape the moment he's done it. Equally, this beautiful woman who is an innocent, why would she decide to do this? What would be in her that she would first be so distraught and then calmly decide to kill herself?"

O'Sullivan's task now is to recommit the entire thing to memory: "I learn six pages a night before I go to sleep," she says. "I'm swimming in Shakespeare. It's like when people start learning a foreign language and end up dreaming in French – I am dreaming in Shakespeare."

The experience, she says, is changing everything. "I come from a land of singer-songwriters, and in Ireland it can seem a bit below par if you're not singing your own stuff. It's only recently that I've seen that what I do is much more like what an actor does. This has opened up a whole new world of literature, of poems, of novels. We've been thinking, 'What about doing Beckett?'"